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First Order Properties
Philippe Darondeau, Stephane Demri, Roland Meyer, Christophe Morvan
To cite this version:
Philippe Darondeau, Stephane Demri, Roland Meyer, Christophe Morvan. Petri Net Reachability
Graphs: Decidability Status of First Order Properties. Logical Methods in Computer Science, Logical
Methods in Computer Science Association, 2012, 8 (4:9), pp.1-28. �10.2168/LMCS-8(4:9)2012�. �hal-
00743935�
PETRI NET REACHABILITY GRAPHS:
DECIDABILITY STATUS OF FIRST-ORDER PROPERTIES ∗
PHILIPPE DARONDEAU
a, STEPHANE DEMRI
b, ROLAND MEYER
c, AND CHRISTOPHE MORVAN
da
INRIA Rennes Bretagne Atlantique, Campus de Beaulieu, Rennes, France e-mail address: philippe.darondeau@inria.fr
b
LSV, ENS Cachan, CNRS, INRIA, France e-mail address: Stephane.Demri@lsv.ens-cachan.fr
c
University of Kaiserslautern, Germany e-mail address: meyer@cs.uni-kl.de
d
Universit´e Paris-Est, Marne-La-Vall´ee, France e-mail address: christophe.morvan@univ-paris-est.fr
Abstract. We investigate the decidability and complexity status of model-checking prob- lems on unlabelled reachability graphs of Petri nets by considering first-order and modal languages without labels on transitions or atomic propositions on markings. We consider several parameters to separate decidable problems from undecidable ones. Not only are we able to provide precise borders and a systematic analysis, but we also demonstrate the robustness of our proof techniques.
1. Introduction
Decision problems for Petri nets. Petri nets are among the oldest families of generators of infinite state systems, and much effort has been dedicated to their algorithmic analysis. For Petri nets, the reachability problem is hard but decidable [35]. Further important prob- lems that are specific to Petri nets and that were shown decidable are boundedness [29, 38], deadlock-freeness and liveness [20] (by reduction to reachability), persistence [18], and semi- linearity [22]. Hack’s thesis [20] provides a comprehensive overview of problems equivalent to Petri net reachability. On the negative side, language equality is undecidable for labelled Petri nets [21, 1], but it can be decided for injectively labelled as well as for labelled and deterministic Petri nets [37] (by a reduction to reachability). Another undecidability result for Petri nets, obtained by Rabin [4] and Hack [21], is that equality of reachability sets of two Petri nets with identical places is undecidable. As our main contribution, we link this result to first-order logic expressing properties of general Petri net reachability graphs.
1998 ACM Subject Classification: F.1.1, F.4.1.
Key words and phrases: Petri nets, First order logic, Reachability graph.
∗
A short version of this paper was published at FSTTCS 2011.
LOGICAL METHODS
lIN COMPUTER SCIENCE DOI:10.2168/LMCS-8(4:9)2012
c P. Darondeau, S. Demri, R. Meyer, and C. Morvan CC Creative Commons